The potential move, which was reported by Bloomberg, would come after the company dropped co-branding with American Express in Canada earlier this year. The retailer is now soliciting bids for an issuer and payments network for U.S. cards, the report said, and American Express could well be among these bids.

In January, Costco's former CEO said the single-card arrangement helps maintain security against the sort of breaches suffered by other retailers.

"Obviously, I'm sure that Target and Neiman Marcus had taken the precautions that were necessary," Costco co-founder James Sinegal told CNBC. "We have a limited exposure to credit cards because we only accept American Express at our warehouses."